Abstract
As one of the demand-side programs, time-of-use (TOU) tariffs brings opportunities of maintaining power grid stability for electricity providers and chances of energy conservation for manufacturers, but it also brings challenge for enterprises to optimize scheduling schemes. This paper studies a two-stage parallel machine scheduling problem under TOU to minimize total electricity costs. The two-stage parallel machine system is composed of identical parallel speed-scaling machines at stage 1 and unrelated parallel machines at stage 2. The key issues lie in assigning a group of jobs to a set of parallel machines at each stage and choosing the appropriate processing speed for all jobs at stage 1, and then determining the interval of processing time for jobs on each selected machine. To solve this problem, a new continuous-time mixed-integer linear programming model is formulated. According to the characteristics of this model, a tabu search-greedy insertion hybrid (TS-GIH) algorithm is designed, which realizes job-machine assignment based on load balancing principle, job insertion with greedy mechanism as well as movement and speed adjustment strategies to find more suitable positions for jobs. The effectiveness of the proposed TS-GIH is demonstrated by comparing with CLPEX and improved genetic algorithm (IGA) through real-life and randomly generated instances. The results show that TS-GIH can realize the trade-off between computation time and solution quality. Compared with CLPEX, the computation time of TS-GIH is significantly less, and the solution quality is much better than IGA.
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References
Afzalirad, M., & Shafipour, M. (2018). Design of an efficient genetic algorithm for resource-constrained unrelated parallel machine scheduling problem with machine eligibility restrictions. Journal of Intelligent Manufacturing, 29(2), 423–437. https://doi.org/10.1007/s10845-015-1117-6.
Albadi, M. H., & El-Saadany, E. F. (2008). A summary of demand response in electricity markets. Electric Power Systems Research, 78(11), 1989–1996. https://doi.org/10.1016/j.epsr.2008.04.002.
Chao, L., Liang, G., Li, X., Pan, Q., & Qi, W. (2017). Energy-efficient permutation flow shop scheduling problem using a hybrid multi-objective backtracking search algorithm. Journal of Cleaner Production, 144, 228–238. https://doi.org/10.1016/j.jclepro.2017.01.011.
Che, A., Zeng, Y. Z., & Lyu, K. (2016). An efficient greedy insertion heuristic for energy-conscious single machine scheduling problem under time-of-use electricity tariffs. Journal of Cleaner Production, 129, 565–577. https://doi.org/10.1016/j.jclepro.2016.03.150.
Che, A., Zhang, S. B. H., & Wu, X. Q. (2017). Energy-conscious unrelated parallel machine scheduling under time-of-use electricity tariffs. Journal of Cleaner Production, 156, 688–697. https://doi.org/10.1016/j.jclepro.2017.04.018.
Chen, J., Pan, Q. K., Wang, L., & Li, J. Q. (2012). A hybrid dynamic harmony search algorithm for identical parallel machines scheduling. Engineering Optimization, 44(2), 209–224. https://doi.org/10.1080/0305215x.2011.576759.
Cheng, J. H., Chu, F., Liu, M., Wu, P., & Xia, W. L. (2017). Bi-criteria single-machine batch scheduling with machine on/off switching under time-of-use tariffs. Computers & Industrial Engineering, 112, 721–734. https://doi.org/10.1016/j.cie.2017.04.026.
Davis, E., & Jaffe, J. M. (1981). Algorithms for scheduling tasks on unrelated processors. Journal of the Association for Computing Machinery, 28(4), 721–736. https://doi.org/10.1145/322276.322284.
Ding, J. Y., Song, S. J., Zhang, R., Chiong, R., & Wu, C. (2016). Parallel machine scheduling under time-of-use electricity prices: New models and optimization approaches. IEEE Transactions on Automation Science and Engineering, 13(2), 1138–1154. https://doi.org/10.1109/Tase.2015.2495328.
Dong, J., Tong, W., Luo, T., Wang, X., Hu, J., Xu, Y., et al. (2017). An FPTAS for the parallel two-stage flowshop problem. Theoretical Computer Science, 657, 64–72. https://doi.org/10.1016/j.tcs.2016.04.046.
Glover, F. (1986). Future paths for integer programming and links to artificial intelligence. Computers & Operations Research, 13(5), 533–549. https://doi.org/10.1016/0305-0548(86)90048-1.
Godina, R., Rodrigues, E., Pouresmaeil, E., Matias, J., & Catalão, J. (2018). Model predictive control home energy management and optimization strategy with demand response. Applied Sciences, 8(3), 408. https://doi.org/10.3390/app8030408.
International Energy Agency (IEA). (2019). World Energy Outlook 2018: The gold standard of energy analysis. Retrieved May 15, 2019 from https://www.iea.org/weo2018/.
Jin, X. B., Zhang, F., Fan, L. Y., Song, Y., & Liu, Z. Y. (2015). Scheduling for energy minimization on restricted parallel processors. Journal of Parallel and Distributed Computing, 81–82, 36–46. https://doi.org/10.1016/j.jpdc.2015.04.001.
Jovane, F., Yoshikawa, H., Alting, L., Boër, C. R., Westkamper, E., Williams, D., et al. (2008). The incoming global technological and industrial revolution towards competitive sustainable manufacturing. CIRP Annals—Manufacturing Technology, 57(2), 641–659. https://doi.org/10.1016/j.cirp.2008.09.010.
Kan, F., Uhan, N. A., Fu, Z., & Sutherland, J. W. (2016). Scheduling on a single machine under time-of-use electricity tariffs. Annals of Operations Research, 238(1–2), 199–227. https://doi.org/10.1007/s10479-015-2003-5.
Lin, C., Luo, W., & Zhang, G. (2011). Approximation algorithms for unrelated machine scheduling with an energy budget. In Frontiers in algorithmics and algorithmic aspects in information and management—Joint international conference, FAW-AAIM 2011, Jinhua, China, May 28–31, 2011. Proceedings. http://doi.org/10.1007/978-3-642-21204-8_27.
Luo, H., Du, B., Huang, G. Q., Chen, H., & Li, X. (2013). Hybrid flow shop scheduling considering machine electricity consumption cost. International Journal of Production Economics, 146(2), 423–439. https://doi.org/10.1016/j.ijpe.2013.01.028.
Moon, J. Y., & Park, J. (2014). Smart production scheduling with time-dependent and machine-dependent electricity cost by considering distributed energy resources and energy storage. International Journal of Production Research, 52(13), 3922–3939. https://doi.org/10.1080/00207543.2013.860251.
Moon, J. Y., Shin, K., & Park, J. (2013). Optimization of production scheduling with time-dependent and machine-dependent electricity cost for industrial energy efficiency. International Journal of Advanced Manufacturing Technology, 68(1–4), 523–535. https://doi.org/10.1007/s00170-013-4749-8.
Pinedo, M. (2012). Scheduling: Theory, algorithms, and systems. New York: Springer.
Qin, W., Zhang, J., & Song, D. (2018). An improved ant colony algorithm for dynamic hybrid flow shop scheduling with uncertain processing time. Journal of Intelligent Manufacturing, 29(4), 891–904. https://doi.org/10.1007/s10845-015-1144-3.
Ramezani, P., Rabiee, M., & Jolai, F. (2015). No-wait flexible flowshop with uniform parallel machines and sequence-dependent setup time: A hybrid meta-heuristic approach. Journal of Intelligent Manufacturing, 26(4), 731–744. https://doi.org/10.1007/s10845-013-0830-2.
Shim, S. O., & Kim, Y. D. (2007). Minimizing total tardiness in an unrelated parallel-machine scheduling problem. Journal of the Operational Research Society, 58(3), 346–354. https://doi.org/10.1057/palgrave.jors.2602141.
Tan, M., Yang, H. L., Duan, B., Su, Y. X., & He, F. (2017). Optimizing production scheduling of steel plate hot rolling for economic load dispatch under time-of-use electricity pricing. Mathematical Problems in Engineering, 2017, 1–13. https://doi.org/10.1155/2017/1048081.
Toksarı, M. D., & Güner, E. (2010). Parallel machine scheduling problem to minimize the earliness/tardiness costs with learning effect and deteriorating jobs. Journal of Intelligent Manufacturing, 21(6), 843–851. https://doi.org/10.1007/s10845-009-0260-3.
Wang, Y., & Li, L. (2013). Time-of-use based electricity demand response for sustainable manufacturing systems. Energy, 63(1), 233–244. https://doi.org/10.1016/j.energy.2013.10.011.
Wang, S. J., Wang, X. D., Yu, J. B., Ma, S., & Liu, M. (2018). Bi-objective identical parallel machine scheduling to minimize total energy consumption and makespan. Journal of Cleaner Production, 193, 424–440. https://doi.org/10.1016/j.jclepro.2018.05.056.
Wang, S., Zhu, Z., Kan, F., Feng, C., & Chu, C. (2017). Scheduling on a two-machine permutation flow shop under time-of-use electricity tariffs. International Journal of Production Research, 56(9), 3173–3187. https://doi.org/10.1080/00207543.2017.1401236.
Wu, X. Q., & Che, A. (2019). A memetic differential evolution algorithm for energy-efficient parallel machine scheduling. Omega-International Journal of Management Science, 82, 155–165. https://doi.org/10.1016/j.omega.2018.01.001.
Yao, F., Demers, A., & Shenker, S. (1995). A scheduling model for reduced CPU energy. In Proceedings of IEEE 36th annual foundations of computer science, Milwaukee, WI, USA, 23–25 Oct 1995. http://doi.org/10.1109/SFCS.1995.492493.
Zeng, Y. Z., Che, A., & Wu, X. (2017). Bi-objective scheduling on uniform parallel machines considering electricity cost. Engineering Optimization, 50(1), 19–36. https://doi.org/10.1080/0305215X.2017.1296437.
Zeng, Y. J., & Sun, Y. G. (2015). Short-term scheduling of steam power system in iron and steel industry under time-of-use power price. Journal of Iron and Steel Research International, 22(9), 795–803. https://doi.org/10.1016/S1006-706x(15)30073-X.
Zhang, H., Dai, Z., Zhang, W., Zhang, S., Wang, Y., & Liu, R. (2017). A new energy-aware flexible job shop scheduling method using modified biogeography-based optimization. Mathematical Problems in Engineering, 2017, 1–12. https://doi.org/10.1155/2017/7249876.
Zhang, G., Gao, L., & Shi, Y. (2011). An effective genetic algorithm for the flexible job-shop scheduling problem. Expert Systems with Applications, 38(4), 3563–3573.
Zhang, H., Zhao, F., Fang, K., & Sutherland, J. W. (2014). Energy-conscious flow shop scheduling under time-of-use electricity tariffs. CIRP Annals—Manufacturing. Technology, 63(1), 37–40. https://doi.org/10.1016/j.cirp.2014.03.011.
Zhao, S., Grossmann, I. E., & Tang, L. (2017). Integrated scheduling of rolling sector in steel production with consideration of energy consumption under time-of-use electricity prices. Computers & Chemical Engineering, 111, 55–65. https://doi.org/10.1016/j.compchemeng.2017.12.018.
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This work was supported by the National Natural Science Foundation of China (Grant No. 71772002).
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Zhang, H., Wu, Y., Pan, R. et al. Two-stage parallel speed-scaling machine scheduling under time-of-use tariffs. J Intell Manuf 32, 91–112 (2021). https://doi.org/10.1007/s10845-020-01561-6
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DOI: https://doi.org/10.1007/s10845-020-01561-6